<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmoothIntegers</code>( <var class="Arg">maxp</var>, <var class="Arg">maxn</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmoothIntegers</code>( <var class="Arg">L</var>, <var class="Arg">maxp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>
<p>The function <code class="code">AllSmoothIntegers(<var class="Arg">maxp</var>,<var class="Arg">maxn</var>)</code> returns the list of all positive integers less than or equal to <var class="Arg">maxn</var> whose prime factors are all in the list <span class="SimpleMath">\(L = \{p ~|~ p \leqslant maxp, p~\mbox{prime} \}\)</span>.</p>
<p>In the alternative form, when <span class="SimpleMath">\(L\)</span> is a list of primes, the function returns the list of all positive integers whose prime factors lie in <span class="SimpleMath">\(L\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllProducts</code>( <var class="Arg">L</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>
<p>The command <code class="code">AllProducts(<var class="Arg">L</var>,<var class="Arg">k</var>)</code> returns the list of all products of <var class="Arg">k</var> entries of the list <var class="Arg">L</var>. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedPartitionsWithoutRepetitions</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>
<p>For a positive integer <var class="Arg">n</var> and a set of positive integers <var class="Arg">S</var>, this function returns the list of partitions of <var class="Arg">n</var> into distinct elements of <var class="Arg">S</var>. Unlike <code class="code">RestrictedPartitions</code>, no repetitions are allowed.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NextProbablyPrimeInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>
<p>The function <code class="code">NextProbablyPrimeInt(<var class="Arg">n</var>)</code> does the same as <code class="code">NextPrimeInt(<var class="Arg">n</var>)</code> except that for reasons of performance it tests numbers only for <code class="code">IsProbablyPrimeInt(<var class="Arg">n</var>)</code> instead of <code class="code">IsPrimeInt(<var class="Arg">n</var>)</code>. For large <var class="Arg">n</var>, this function is much faster than <code class="code">NextPrimeInt(<var class="Arg">n</var>)</code></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimeNumbersIterator</code>( [<var class="Arg">chunksize</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function has been transferred from package <strong class="pkg">RCWA</strong>.</p>
<p>This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument <code class="code">chunksize</code> which specifies the length of the interval which is sieved in one go (the default is <span class="SimpleMath">\(10^7\)</span>), and which can be used to balance runtime vs. memory consumption. It is assumed that <code class="code">chunksize</code> is larger than any gap between two consecutive primes within the range one intends to run the iterator over.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter := PrimeNumbersIterator();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..100] do p := NextIterator(iter); od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p;</span>
541
<span class="GAPprompt">gap></span> <span class="GAPinput">sum := 0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">## "prime number race" 1 vs. 3 mod 4</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for p in PrimeNumbersIterator() do </span>
<span class="GAPprompt">></span> <span class="GAPinput"> if p <> 2 then sum := sum + E(4)^(p-1); fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if sum > 0 then break; fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p;</span>
26861
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